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The Localization-Topology Correspondence:
Periodic Systems and Beyond
Gianluca Panati
Dipartimento di Matematica ”G. Castelnuovo”
based on joint work with G. Marcelli, D. Monaco,
M. Moscolari, A. Pisante, S. Teufel
Mathematical Physics of Anyons and TopologicalStates of Matter
NORDITA Stockholm, March 2019
Plan of the talk
� Intro: from the Transport-Topology Correspondence [TKNN]to the Localization-Topology Correspondence
I. Chern insulators vs ordinary insulators: the periodic casebased on joint paper with D. Monaco, A. Pisante and S. Teufel(Commun. Math. Phys. 359 (2018))
II. Chern insulators vs ordinary insulators: the non-periodic casework in progress with G. Marcelli and M. Moscolari(preliminary results in one direction, conjecture in the other direction)
G. Panati La Sapienza The Localization-Topology Correspondence 2 / 33
Plan of the talk
� Intro: from the Transport-Topology Correspondence [TKNN]to the Localization-Topology Correspondence
I. Chern insulators vs ordinary insulators: the periodic casebased on joint paper with D. Monaco, A. Pisante and S. Teufel(Commun. Math. Phys. 359 (2018))
II. Chern insulators vs ordinary insulators: the non-periodic casework in progress with G. Marcelli and M. Moscolari(preliminary results in one direction, conjecture in the other direction)
G. Panati La Sapienza The Localization-Topology Correspondence 2 / 33
Plan of the talk
� Intro: from the Transport-Topology Correspondence [TKNN]to the Localization-Topology Correspondence
I. Chern insulators vs ordinary insulators: the periodic casebased on joint paper with D. Monaco, A. Pisante and S. Teufel(Commun. Math. Phys. 359 (2018))
II. Chern insulators vs ordinary insulators: the non-periodic casework in progress with G. Marcelli and M. Moscolari(preliminary results in one direction, conjecture in the other direction)
G. Panati La Sapienza The Localization-Topology Correspondence 2 / 33
The predecessor: Transport-Topology Correspondence
I Idea originated in [TKNN 82] in the context of the Quantum Hall effect.
I Retrospectively, one can rephrase the idea as follows:
periodic* HamiltonianBF−→ Fermi projector {P(k)}k∈Td −→ Bloch bundle EP
σ(Kubo)xy︸ ︷︷ ︸
Transport
= − i
(2π)2
∫T2
Tr(P(k)[∂k1P(k), ∂k2P(k)]
)dk =
1
2πC1(EP)︸ ︷︷ ︸Topology
I Math results & generalizations: [AS2, AG, Be, BES, BGKS, Gr, ES, . . . ].See also [AW] for the role of the Linear Response Ansatz.
I Recent generalizations to interacting electrons [GMP; BRF, MT].
I Other complementary explanations of the QHE are possible, but not coveredhere [Laughlin, Frohlich, Halperin]. Some can be adapted to topologicalinsulators.
*) Periodic either with respect to magnetic translations (QHE) or with respect to
ordinary translations (Chern insulators)
G. Panati La Sapienza The Localization-Topology Correspondence 3 / 33
The predecessor: Transport-Topology Correspondence
I Idea originated in [TKNN 82] in the context of the Quantum Hall effect.
I Retrospectively, one can rephrase the idea as follows:
periodic* HamiltonianBF−→ Fermi projector {P(k)}k∈Td −→ Bloch bundle EP
σ(Kubo)xy︸ ︷︷ ︸
Transport
= − i
(2π)2
∫T2
Tr(P(k)[∂k1P(k), ∂k2P(k)]
)dk =
1
2πC1(EP)︸ ︷︷ ︸Topology
I Math results & generalizations: [AS2, AG, Be, BES, BGKS, Gr, ES, . . . ].See also [AW] for the role of the Linear Response Ansatz.
I Recent generalizations to interacting electrons [GMP; BRF, MT].
I Other complementary explanations of the QHE are possible, but not coveredhere [Laughlin, Frohlich, Halperin]. Some can be adapted to topologicalinsulators.
*) Periodic either with respect to magnetic translations (QHE) or with respect to
ordinary translations (Chern insulators)
G. Panati La Sapienza The Localization-Topology Correspondence 3 / 33
The predecessor: Transport-Topology Correspondence
I Idea originated in [TKNN 82] in the context of the Quantum Hall effect.
I Retrospectively, one can rephrase the idea as follows:
periodic* HamiltonianBF−→ Fermi projector {P(k)}k∈Td −→ Bloch bundle EP
σ(Kubo)xy︸ ︷︷ ︸
Transport
= − i
(2π)2
∫T2
Tr(P(k)[∂k1P(k), ∂k2P(k)]
)dk =
1
2πC1(EP)︸ ︷︷ ︸Topology
I Math results & generalizations: [AS2, AG, Be, BES, BGKS, Gr, ES, . . . ].See also [AW] for the role of the Linear Response Ansatz.
I Recent generalizations to interacting electrons [GMP; BRF, MT].
I Other complementary explanations of the QHE are possible, but not coveredhere [Laughlin, Frohlich, Halperin]. Some can be adapted to topologicalinsulators.
*) Periodic either with respect to magnetic translations (QHE) or with respect to
ordinary translations (Chern insulators)
G. Panati La Sapienza The Localization-Topology Correspondence 3 / 33
The predecessor: Transport-Topology Correspondence
I Idea originated in [TKNN 82] in the context of the Quantum Hall effect.
I Retrospectively, one can rephrase the idea as follows:
periodic* HamiltonianBF−→ Fermi projector {P(k)}k∈Td −→ Bloch bundle EP
σ(Kubo)xy︸ ︷︷ ︸
Transport
= − i
(2π)2
∫T2
Tr(P(k)[∂k1P(k), ∂k2P(k)]
)dk =
1
2πC1(EP)︸ ︷︷ ︸Topology
I Math results & generalizations: [AS2, AG, Be, BES, BGKS, Gr, ES, . . . ].See also [AW] for the role of the Linear Response Ansatz.
I Recent generalizations to interacting electrons [GMP; BRF, MT].
I Other complementary explanations of the QHE are possible, but not coveredhere [Laughlin, Frohlich, Halperin]. Some can be adapted to topologicalinsulators.
*) Periodic either with respect to magnetic translations (QHE) or with respect to
ordinary translations (Chern insulators)
G. Panati La Sapienza The Localization-Topology Correspondence 3 / 33
The predecessor: Transport-Topology Correspondence
I Idea originated in [TKNN 82] in the context of the Quantum Hall effect.
I Retrospectively, one can rephrase the idea as follows:
periodic* HamiltonianBF−→ Fermi projector {P(k)}k∈Td −→ Bloch bundle EP
σ(Kubo)xy︸ ︷︷ ︸
Transport
= − i
(2π)2
∫T2
Tr(P(k)[∂k1P(k), ∂k2P(k)]
)dk =
1
2πC1(EP)︸ ︷︷ ︸Topology
I Math results & generalizations: [AS2, AG, Be, BES, BGKS, Gr, ES, . . . ].See also [AW] for the role of the Linear Response Ansatz.
I Recent generalizations to interacting electrons [GMP; BRF, MT].
I Other complementary explanations of the QHE are possible, but not coveredhere [Laughlin, Frohlich, Halperin]. Some can be adapted to topologicalinsulators.
*) Periodic either with respect to magnetic translations (QHE) or with respect to
ordinary translations (Chern insulators)
G. Panati La Sapienza The Localization-Topology Correspondence 3 / 33
The predecessor: Transport-Topology Correspondence
G. Panati La Sapienza The Localization-Topology Correspondence 4 / 33
Chern insulators and the Quantum Anomalous Hall effect
Left panel: transverse and direct resistivity in a Quantum Hall experimentRight panel: transverse and direct resistivity in a Chern insulator (histeresis cycle)
Picture: c© A. J. Bestwick (2015)
G. Panati La Sapienza The Localization-Topology Correspondence 5 / 33
G. Panati La Sapienza The Localization-Topology Correspondence 6 / 33
Part I
The Localization-Topology Correspondence:
the periodic case
Joint work with D. Monaco, A. Pisante and S. Teufel
G. Panati La Sapienza The Localization-Topology Correspondence 7 / 33
How to measure localization of extended states?
I Our question: Relation between dissipationless transport, topology of theBloch bundle and localization of electronic states.
What is the relevantnotion of localizationfor periodic quantum
systems?
I Spectral type?? For ergodic random Schrodinger operators localization ismeasured by the spectral type (σpp, σac, σsc) [AW]. However, for periodicsystems the spectrum is generically purely absolutely continuous.
I Kernel of the Fermi projector?? For gapped periodic 1-body Hamiltoniansone has
|Pµ(x , y)| ' e−λgap|x−y |
as a consequence of Combes-Thomas theory [AS2, NN].
[AW] M. Aizenman, S. Warzel: Random operators, AMS (2015).[AS2] J. Avron, R. Seiler, B. Simon: Commun. Math. Phys. 159 (1994).[NN] A. Nenciu, G. Nenciu: Phys. Rev B 47 (1993).
G. Panati La Sapienza The Localization-Topology Correspondence 8 / 33
How to measure localization of extended states?
I Our question: Relation between dissipationless transport, topology of theBloch bundle and localization of electronic states.
What is the relevantnotion of localizationfor periodic quantum
systems?
I Spectral type?? For ergodic random Schrodinger operators localization ismeasured by the spectral type (σpp, σac, σsc) [AW]. However, for periodicsystems the spectrum is generically purely absolutely continuous.
I Kernel of the Fermi projector?? For gapped periodic 1-body Hamiltoniansone has
|Pµ(x , y)| ' e−λgap|x−y |
as a consequence of Combes-Thomas theory [AS2, NN].
[AW] M. Aizenman, S. Warzel: Random operators, AMS (2015).[AS2] J. Avron, R. Seiler, B. Simon: Commun. Math. Phys. 159 (1994).[NN] A. Nenciu, G. Nenciu: Phys. Rev B 47 (1993).
G. Panati La Sapienza The Localization-Topology Correspondence 8 / 33
How to measure localization of extended states?
I Our question: Relation between dissipationless transport, topology of theBloch bundle and localization of electronic states.
What is the relevantnotion of localizationfor periodic quantum
systems?
I Spectral type?? For ergodic random Schrodinger operators localization ismeasured by the spectral type (σpp, σac, σsc) [AW]. However, for periodicsystems the spectrum is generically purely absolutely continuous.
I Kernel of the Fermi projector?? For gapped periodic 1-body Hamiltoniansone has
|Pµ(x , y)| ' e−λgap|x−y |
as a consequence of Combes-Thomas theory [AS2, NN].
[AW] M. Aizenman, S. Warzel: Random operators, AMS (2015).[AS2] J. Avron, R. Seiler, B. Simon: Commun. Math. Phys. 159 (1994).[NN] A. Nenciu, G. Nenciu: Phys. Rev B 47 (1993).
G. Panati La Sapienza The Localization-Topology Correspondence 8 / 33
How to measure localization of extended states?
I Our question: Relation between dissipationless transport, topology of theBloch bundle and localization of electronic states.
What is the relevantnotion of localizationfor periodic quantum
systems?
I Spectral type?? For ergodic random Schrodinger operators localization ismeasured by the spectral type (σpp, σac, σsc) [AW]. However, for periodicsystems the spectrum is generically purely absolutely continuous.
I Kernel of the Fermi projector?? For gapped periodic 1-body Hamiltoniansone has
|Pµ(x , y)| ' e−λgap|x−y |
as a consequence of Combes-Thomas theory [AS2, NN].
[AW] M. Aizenman, S. Warzel: Random operators, AMS (2015).[AS2] J. Avron, R. Seiler, B. Simon: Commun. Math. Phys. 159 (1994).[NN] A. Nenciu, G. Nenciu: Phys. Rev B 47 (1993).
G. Panati La Sapienza The Localization-Topology Correspondence 8 / 33
How to measure localization of extended states?
I Our question: Relation between dissipationless transport, topology of theBloch bundle and localization of electronic states.
What is the relevantnotion of localizationfor periodic quantum
systems?
I Spectral type?? For ergodic random Schrodinger operators localization ismeasured by the spectral type (σpp, σac, σsc) [AW]. However, for periodicsystems the spectrum is generically purely absolutely continuous.
I Kernel of the Fermi projector?? For gapped periodic 1-body Hamiltoniansone has
|Pµ(x , y)| ' e−λgap|x−y |
as a consequence of Combes-Thomas theory [AS2, NN].
[AW] M. Aizenman, S. Warzel: Random operators, AMS (2015).[AS2] J. Avron, R. Seiler, B. Simon: Commun. Math. Phys. 159 (1994).[NN] A. Nenciu, G. Nenciu: Phys. Rev B 47 (1993).
G. Panati La Sapienza The Localization-Topology Correspondence 8 / 33
The localization of electrons isconveniently expressed by
Composite Wannier functions (CWFs)
The Fermi projector Pµ reads, for {wa,γ}γ∈Γ,1≤a≤m a system of CWFs,
Pµ =m∑
a=1
∑γ∈Γ
|wa,γ〉 〈wa,γ | .
Notice that crucially
|Pµ(x , y)| ' e−λgap|x−y | ;
{exist CWFs such that
|wa,γ(x)| ' e−c|x−γ|
(GAP CONDITION) ; (TOPOLOGICAL TRIVIALITY)
G. Panati La Sapienza The Localization-Topology Correspondence 9 / 33
The localization of electrons isconveniently expressed by
Composite Wannier functions (CWFs)
The Fermi projector Pµ reads, for {wa,γ}γ∈Γ,1≤a≤m a system of CWFs,
Pµ =m∑
a=1
∑γ∈Γ
|wa,γ〉 〈wa,γ | .
Notice that crucially
|Pµ(x , y)| ' e−λgap|x−y | ;
{exist CWFs such that
|wa,γ(x)| ' e−c|x−γ|
(GAP CONDITION) ; (TOPOLOGICAL TRIVIALITY)
G. Panati La Sapienza The Localization-Topology Correspondence 9 / 33
The localization of electrons isconveniently expressed by
Composite Wannier functions (CWFs)
The Fermi projector Pµ reads, for {wa,γ}γ∈Γ,1≤a≤m a system of CWFs,
Pµ =m∑
a=1
∑γ∈Γ
|wa,γ〉 〈wa,γ | .
Notice that crucially
|Pµ(x , y)| ' e−λgap|x−y | ;
{exist CWFs such that
|wa,γ(x)| ' e−c|x−γ|
(GAP CONDITION) ; (TOPOLOGICAL TRIVIALITY)
G. Panati La Sapienza The Localization-Topology Correspondence 9 / 33
The localization of electrons isconveniently expressed by
Composite Wannier functions (CWFs)
The Fermi projector Pµ reads, for {wa,γ}γ∈Γ,1≤a≤m a system of CWFs,
Pµ =m∑
a=1
∑γ∈Γ
|wa,γ〉 〈wa,γ | .
Notice that crucially
|Pµ(x , y)| ' e−λgap|x−y | ;
{exist CWFs such that
|wa,γ(x)| ' e−c|x−γ|
(GAP CONDITION) ; (TOPOLOGICAL TRIVIALITY)
G. Panati La Sapienza The Localization-Topology Correspondence 9 / 33
Theorem [MPPT]: The Localization Dichotomy for periodic systems
Under assumptions specified later, the following holds:
I either there exists α > 0 and a choice of CWFs w = (w1, . . . , wm) satisfying
m∑a=1
∫Rd
e2β|x| |wa(x)|2dx < +∞ for every β ∈ [0, α);
I or for every possible choice of CWFs w = (w1, . . . ,wm) one has
〈X 2〉w =m∑
a=1
∫Rd
|x |2 |wa(x)|2dx = +∞.
Intermediate regimes areforbidden!!
The result is largelymodel-independent: it
holds for tight-binding aswell as continuum models
[MPPT] Monaco, Panati, Pisante, Teufel: Commun. Math. Phys. 359 (2018).
G. Panati La Sapienza The Localization-Topology Correspondence 10 / 33
Theorem [MPPT]: The Localization Dichotomy for periodic systems
Under assumptions specified later, the following holds:
I either there exists α > 0 and a choice of CWFs w = (w1, . . . , wm) satisfying
m∑a=1
∫Rd
e2β|x| |wa(x)|2dx < +∞ for every β ∈ [0, α);
I or for every possible choice of CWFs w = (w1, . . . ,wm) one has
〈X 2〉w =m∑
a=1
∫Rd
|x |2 |wa(x)|2dx = +∞.
Intermediate regimes areforbidden!!
The result is largelymodel-independent: it
holds for tight-binding aswell as continuum models
[MPPT] Monaco, Panati, Pisante, Teufel: Commun. Math. Phys. 359 (2018).
G. Panati La Sapienza The Localization-Topology Correspondence 10 / 33
Theorem [MPPT]: The Localization Dichotomy for periodic systems
Under assumptions specified later, the following holds:
I either there exists α > 0 and a choice of CWFs w = (w1, . . . , wm) satisfying
m∑a=1
∫Rd
e2β|x| |wa(x)|2dx < +∞ for every β ∈ [0, α);
I or for every possible choice of CWFs w = (w1, . . . ,wm) one has
〈X 2〉w =m∑
a=1
∫Rd
|x |2 |wa(x)|2dx = +∞.
Intermediate regimes areforbidden!!
The result is largelymodel-independent: it
holds for tight-binding aswell as continuum models
[MPPT] Monaco, Panati, Pisante, Teufel: Commun. Math. Phys. 359 (2018).
G. Panati La Sapienza The Localization-Topology Correspondence 10 / 33
Theorem [MPPT]: The Localization Dichotomy for periodic systems
Under assumptions specified later, the following holds:
I either there exists α > 0 and a choice of CWFs w = (w1, . . . , wm) satisfying
m∑a=1
∫Rd
e2β|x| |wa(x)|2dx < +∞ for every β ∈ [0, α);
I or for every possible choice of CWFs w = (w1, . . . ,wm) one has
〈X 2〉w =m∑
a=1
∫Rd
|x |2 |wa(x)|2dx = +∞.
Intermediate regimes areforbidden!!
The result is largelymodel-independent: it
holds for tight-binding aswell as continuum models
[MPPT] Monaco, Panati, Pisante, Teufel: Commun. Math. Phys. 359 (2018).
G. Panati La Sapienza The Localization-Topology Correspondence 10 / 33
Theorem [MPPT]: The Localization Dichotomy for periodic systems
Under assumptions specified later, the following holds:
I either there exists α > 0 and a choice of CWFs w = (w1, . . . , wm) satisfying
m∑a=1
∫Rd
e2β|x| |wa(x)|2dx < +∞ for every β ∈ [0, α);
I or for every possible choice of CWFs w = (w1, . . . ,wm) one has
〈X 2〉w =m∑
a=1
∫Rd
|x |2 |wa(x)|2dx = +∞.
Intermediate regimes areforbidden!!
The result is largelymodel-independent: it
holds for tight-binding aswell as continuum models
[MPPT] Monaco, Panati, Pisante, Teufel: Commun. Math. Phys. 359 (2018).
G. Panati La Sapienza The Localization-Topology Correspondence 10 / 33
Synopsis: symmetry, localization, transport, topology
G. Panati La Sapienza The Localization-Topology Correspondence 11 / 33
Setting: Magnetic periodic Schrodinger operators
Chern insulators: For d ∈ {2, 3}, consider the magnetic Schrodinger operator
HΓ = 12
(−i∇x − AΓ(x))2 + VΓ(x) acting in L2(Rd)
where AΓ and VΓ are periodic with respect to Γ = SpanZ {a1, . . . , ad}.
Hence the modified Bloch-Floquet transform (Zak transform)
(U ψ)(k, y) :=∑γ∈Γ
e−ik·(y−γ) (Tγ ψ)(y), y ∈ Rd , k ∈ Rd
provides a simultaneous decomposition of HΓ and Tγ :
UHΓU−1 =
∫ ⊕B
dk H(k) with H(k) = 12
(− i∇y − AΓ(y) + k
)2+ VΓ(y).
The operator H(k) acts in L2per(Rd) :=
{ψ ∈ L2
loc(Rd) : Tγψ = ψ for all γ ∈ Γ}
.
G. Panati La Sapienza The Localization-Topology Correspondence 12 / 33
Setting: Magnetic periodic Schrodinger operators
Chern insulators: For d ∈ {2, 3}, consider the magnetic Schrodinger operator
HΓ = 12
(−i∇x − AΓ(x))2 + VΓ(x) acting in L2(Rd)
where AΓ and VΓ are periodic with respect to Γ = SpanZ {a1, . . . , ad}.
Hence the modified Bloch-Floquet transform (Zak transform)
(U ψ)(k , y) :=∑γ∈Γ
e−ik·(y−γ) (Tγ ψ)(y), y ∈ Rd , k ∈ Rd
provides a simultaneous decomposition of HΓ and Tγ :
UHΓU−1 =
∫ ⊕B
dk H(k) with H(k) = 12
(− i∇y − AΓ(y) + k
)2+ VΓ(y).
The operator H(k) acts in L2per(Rd) :=
{ψ ∈ L2
loc(Rd) : Tγψ = ψ for all γ ∈ Γ}
.
G. Panati La Sapienza The Localization-Topology Correspondence 12 / 33
Setting: Magnetic periodic Schrodinger operators
Chern insulators: For d ∈ {2, 3}, consider the magnetic Schrodinger operator
HΓ = 12
(−i∇x − AΓ(x))2 + VΓ(x) acting in L2(Rd)
where AΓ and VΓ are periodic with respect to Γ = SpanZ {a1, . . . , ad}.
Hence the modified Bloch-Floquet transform (Zak transform)
(U ψ)(k , y) :=∑γ∈Γ
e−ik·(y−γ) (Tγ ψ)(y), y ∈ Rd , k ∈ Rd
provides a simultaneous decomposition of HΓ and Tγ :
UHΓU−1 =
∫ ⊕B
dk H(k) with H(k) = 12
(− i∇y − AΓ(y) + k
)2+ VΓ(y).
The operator H(k) acts in L2per(Rd) :=
{ψ ∈ L2
loc(Rd) : Tγψ = ψ for all γ ∈ Γ}
.
G. Panati La Sapienza The Localization-Topology Correspondence 12 / 33
The Bloch bundle and its topology
For each fixed k ∈ Rd , the operator H(k) has compact resolvent, so pure pointspectrum accumulating at +∞.
Spectrum of H(k) as a function of kj (left).
Spectrum of HΓ =∫⊕ H(k)dk (right).
Question: how can I read from thepicture whether TR-symmetry is broken?
G. Panati La Sapienza The Localization-Topology Correspondence 13 / 33
The Bloch bundle and its topology
For each fixed k ∈ Rd , the operator H(k) has compact resolvent, so pure pointspectrum accumulating at +∞.
An isolated family of J Bloch bands.
Notice that the spectral bands may overlap.
Topology is encoded in the orthogonalprojections on an isolated family of bands
P∗(k) = i2π
∮C∗(k)
(H(k)− z1)−1dz
=∑
n∈I∗ |un(k, ·)〉 〈un(k, ·)| .
where C∗(k) intersects the real line inE−(k) and E+(k) (GAP CONDITION).
G. Panati La Sapienza The Localization-Topology Correspondence 14 / 33
The Bloch bundle and its topology
For each fixed k ∈ Rd , the operator H(k) has compact resolvent, so pure pointspectrum accumulating at +∞.
An isolated family of J Bloch bands.
Notice that the spectral bands may overlap.
Topology is encoded in the orthogonalprojections on an isolated family of bands
P∗(k) = i2π
∮C∗(k)
(H(k)− z1)−1dz
=∑
n∈I∗ |un(k, ·)〉 〈un(k, ·)| .
where C∗(k) intersects the real line inE−(k) and E+(k) (GAP CONDITION).
G. Panati La Sapienza The Localization-Topology Correspondence 14 / 33
A model for Quantum Hall insulators
Quantum Hall insulators: For d ∈ {2, 3}, consider Ab(x) = b2c x ∧ e with e a unit
vectorHΓ,b = 1
2(−i∇x − Ab(x))2 + VΓ(x) acting in L2(Rd)
where VΓ is periodic with respect to Γ = SpanZ {a1, . . . , ad}.Assume moreover the commensurability condition for the magnetic field B = be:
1
cB · (γ ∧ γ′) ∈ 2πQ for all γ, γ′ ∈ Γ
Ordinary translations are replaced by the magnetic translations [Zak64]
(T bγψ)(x) := eiγ·Ab(x) ψ(x − γ) γ ∈ Γ.
These commute with HΓ,b, but satisfy the pseudo-Weyl relations
T bγT
bγ′ = e
ic B·(γ∧γ
′) T bγ′T
bγ γ, γ′ ∈ Γ.
The Zd -symmetry is recovered at the price of considering a smaller lattice Γb ⊂ Γ.
G. Panati La Sapienza The Localization-Topology Correspondence 15 / 33
Assumptions
Consider A = AΓ + Ab (periodic + linear) with Ab satisfying the commensurabilitycondition, and set
H(κ) =1
2
(− i∇y − A(y) + κ
)2+ VΓ(y) acting in Hb
f , κ ∈ Cd ,
where Hbf :=
{ψ ∈ L2
loc(Rd) : T bγψ = ψ, for all γ ∈ Γb
}.
Assumption 1: The magnetic potential A = AΓ + Ab and the scalar potential VΓ
are such that the family of operators H(κ) is an entire analytic family in the senseof Kato with compact resolvent.
If A = AΓ is Γ-periodic, with fundamental unit cell Y , then it is sufficient to assume either:
A ∈ L∞(Y ;R2) when d = 2 or A ∈ L4(Y ;R3) when d = 3, and divA,VΓ ∈ L2loc(Rd ) when
2 ≤ d ≤ 3;
A ∈ Lr (Y ;R2) with r > 2 and VΓ ∈ Lp(Y ) with p > 1 when d = 2, or A ∈ L3(Y ;R3) andVΓ ∈ L3/2(Y ) when d = 3 (compare [BS]).
[BS] Birman, Suslina: Algebra i Analiz 11 (1999). St. Petersburg Math. J. 11, (2000).
G. Panati La Sapienza The Localization-Topology Correspondence 16 / 33
Assumptions
Consider A = AΓ + Ab (periodic + linear) with Ab satisfying the commensurabilitycondition, and set
H(κ) =1
2
(− i∇y − A(y) + κ
)2+ VΓ(y) acting in Hb
f , κ ∈ Cd ,
where Hbf :=
{ψ ∈ L2
loc(Rd) : T bγψ = ψ, for all γ ∈ Γb
}.
Assumption 1: The magnetic potential A = AΓ + Ab and the scalar potential VΓ
are such that the family of operators H(κ) is an entire analytic family in the senseof Kato with compact resolvent.
If A = AΓ is Γ-periodic, with fundamental unit cell Y , then it is sufficient to assume either:
A ∈ L∞(Y ;R2) when d = 2 or A ∈ L4(Y ;R3) when d = 3, and divA,VΓ ∈ L2loc(Rd ) when
2 ≤ d ≤ 3;
A ∈ Lr (Y ;R2) with r > 2 and VΓ ∈ Lp(Y ) with p > 1 when d = 2, or A ∈ L3(Y ;R3) andVΓ ∈ L3/2(Y ) when d = 3 (compare [BS]).
[BS] Birman, Suslina: Algebra i Analiz 11 (1999). St. Petersburg Math. J. 11, (2000).
G. Panati La Sapienza The Localization-Topology Correspondence 16 / 33
Lemma: Let P∗(k) be the spectral projector of H(k) corresponding to a set σ∗(k) ⊂R such that the gap condition is satisfied. Then the family {P∗(k)}k∈Rd has thefollowing properties:
(P1) the map k 7→ P∗(k) is analytic from Rd to B(Hbf );
(P2) the map k 7→ P∗(k) is τ -covariant, i. e.
P∗(k + λ) = τ(λ)P∗(k) τ(λ)−1 ∀k ∈ Rd , ∀λ ∈ Γ∗b.
where τ : Γ∗b ' Zd −→ U(Hbf ) is a unitary representation.
Concretely, for the operator HΓ
τ(λ)f (y) = eiλ·y f (y)
for f ∈ L2per(Rd , dy) = H0
f .
In view of (P1) and (P2) the ranges of P∗(k) define a (smooth) Hermitian vectorbundle over Td
∗ := Rd/Γb. For d = 2, it is characterized by the Chern number
c1(P) := 12πi
∫T2∗
TrHbf
(P(k) [∂k1P(k), ∂k2P(k)]
)dk1 ∧ dk2
For the sake of simpler slides, here we consider the case τ ≡ 1.
G. Panati La Sapienza The Localization-Topology Correspondence 17 / 33
Lemma: Let P∗(k) be the spectral projector of H(k) corresponding to a set σ∗(k) ⊂R such that the gap condition is satisfied. Then the family {P∗(k)}k∈Rd has thefollowing properties:
(P1) the map k 7→ P∗(k) is analytic from Rd to B(Hbf );
(P2) the map k 7→ P∗(k) is τ -covariant, i. e.
P∗(k + λ) = τ(λ)P∗(k) τ(λ)−1 ∀k ∈ Rd , ∀λ ∈ Γ∗b.
where τ : Γ∗b ' Zd −→ U(Hbf ) is a unitary representation.
Concretely, for the operator HΓ
τ(λ)f (y) = eiλ·y f (y)
for f ∈ L2per(Rd , dy) = H0
f .
In view of (P1) and (P2) the ranges of P∗(k) define a (smooth) Hermitian vectorbundle over Td
∗ := Rd/Γb. For d = 2, it is characterized by the Chern number
c1(P) := 12πi
∫T2∗
TrHbf
(P(k) [∂k1P(k), ∂k2P(k)]
)dk1 ∧ dk2
For the sake of simpler slides, here we consider the case τ ≡ 1.
G. Panati La Sapienza The Localization-Topology Correspondence 17 / 33
Lemma: Let P∗(k) be the spectral projector of H(k) corresponding to a set σ∗(k) ⊂R such that the gap condition is satisfied. Then the family {P∗(k)}k∈Rd has thefollowing properties:
(P1) the map k 7→ P∗(k) is analytic from Rd to B(Hbf );
(P2) the map k 7→ P∗(k) is τ -covariant, i. e.
P∗(k + λ) = τ(λ)P∗(k) τ(λ)−1 ∀k ∈ Rd , ∀λ ∈ Γ∗b.
where τ : Γ∗b ' Zd −→ U(Hbf ) is a unitary representation.
Concretely, for the operator HΓ
τ(λ)f (y) = eiλ·y f (y)
for f ∈ L2per(Rd , dy) = H0
f .
In view of (P1) and (P2) the ranges of P∗(k) define a (smooth) Hermitian vectorbundle over Td
∗ := Rd/Γb. For d = 2, it is characterized by the Chern number
c1(P) := 12πi
∫T2∗
TrHbf
(P(k) [∂k1P(k), ∂k2P(k)]
)dk1 ∧ dk2
For the sake of simpler slides, here we consider the case τ ≡ 1.
G. Panati La Sapienza The Localization-Topology Correspondence 17 / 33
Lemma: Let P∗(k) be the spectral projector of H(k) corresponding to a set σ∗(k) ⊂R such that the gap condition is satisfied. Then the family {P∗(k)}k∈Rd has thefollowing properties:
(P1) the map k 7→ P∗(k) is analytic from Rd to B(Hbf );
(P2) the map k 7→ P∗(k) is τ -covariant, i. e.
P∗(k + λ) = τ(λ)P∗(k) τ(λ)−1 ∀k ∈ Rd , ∀λ ∈ Γ∗b.
where τ : Γ∗b ' Zd −→ U(Hbf ) is a unitary representation.
Concretely, for the operator HΓ
τ(λ)f (y) = eiλ·y f (y)
for f ∈ L2per(Rd , dy) = H0
f .
In view of (P1) and (P2) the ranges of P∗(k) define a (smooth) Hermitian vectorbundle over Td
∗ := Rd/Γb. For d = 2, it is characterized by the Chern number
c1(P) := 12πi
∫T2∗
TrHbf
(P(k) [∂k1P(k), ∂k2P(k)]
)dk1 ∧ dk2
For the sake of simpler slides, here we consider the case τ ≡ 1.
G. Panati La Sapienza The Localization-Topology Correspondence 17 / 33
Wannier functions and their localization
IDEA: Wannier functions provide a reasonablecompromise between localization in energy and
localization in position space, as far ascompatible with the uncertainty principle.
They are associated at the energy window corresponding to P∗(·) via
Definition: A Bloch frame is a collection {φ1, . . . , φm} ⊂ L2(Td∗ ,Hb
f ) such that:
(φ1(k), . . . , φm(k)) is an orthonormal basis of RanP∗(k) for a.e. k ∈ Rd
In general, a Bloch frame mixes different Bloch bands
φa(k) =∑n∈I∗
un(k)︸ ︷︷ ︸Bloch funct.
Una(k) for some unitary matrix U(k).
The ambiguity in thechoice is dubbed
Bloch gaugeambiguity.
G. Panati La Sapienza The Localization-Topology Correspondence 18 / 33
Wannier functions and their localization
IDEA: Wannier functions provide a reasonablecompromise between localization in energy and
localization in position space, as far ascompatible with the uncertainty principle.
They are associated at the energy window corresponding to P∗(·) via
Definition: A Bloch frame is a collection {φ1, . . . , φm} ⊂ L2(Td∗ ,Hb
f ) such that:
(φ1(k), . . . , φm(k)) is an orthonormal basis of RanP∗(k) for a.e. k ∈ Rd
In general, a Bloch frame mixes different Bloch bands
φa(k) =∑n∈I∗
un(k)︸ ︷︷ ︸Bloch funct.
Una(k) for some unitary matrix U(k).
The ambiguity in thechoice is dubbed
Bloch gaugeambiguity.
G. Panati La Sapienza The Localization-Topology Correspondence 18 / 33
Wannier functions and their localization
IDEA: Wannier functions provide a reasonablecompromise between localization in energy and
localization in position space, as far ascompatible with the uncertainty principle.
They are associated at the energy window corresponding to P∗(·) via
Definition: A Bloch frame is a collection {φ1, . . . , φm} ⊂ L2(Td∗ ,Hb
f ) such that:
(φ1(k), . . . , φm(k)) is an orthonormal basis of RanP∗(k) for a.e. k ∈ Rd
In general, a Bloch frame mixes different Bloch bands
φa(k) =∑n∈I∗
un(k)︸ ︷︷ ︸Bloch funct.
Una(k) for some unitary matrix U(k).
The ambiguity in thechoice is dubbed
Bloch gaugeambiguity.
G. Panati La Sapienza The Localization-Topology Correspondence 18 / 33
Wannier functions and their localization
IDEA: Wannier functions provide a reasonablecompromise between localization in energy and
localization in position space, as far ascompatible with the uncertainty principle.
They are associated at the energy window corresponding to P∗(·) via
Definition: A Bloch frame is a collection {φ1, . . . , φm} ⊂ L2(Td∗ ,Hb
f ) such that:
(φ1(k), . . . , φm(k)) is an orthonormal basis of RanP∗(k) for a.e. k ∈ Rd
In general, a Bloch frame mixes different Bloch bands
φa(k) =∑n∈I∗
un(k)︸ ︷︷ ︸Bloch funct.
Una(k) for some unitary matrix U(k).
The ambiguity in thechoice is dubbed
Bloch gaugeambiguity.
G. Panati La Sapienza The Localization-Topology Correspondence 18 / 33
The Bloch bundle
φ1(k)φ2(k)
k
RanP∗(k) Competition between regularityand periodicity is encoded by theBloch bundle
E = (E → Td∗).
Existence of continuous Blochframes is topologically obstructedif c1(P∗) 6= 0.
G. Panati La Sapienza The Localization-Topology Correspondence 19 / 33
Definition (CWFs): The composite Wannier functions {w1, . . . ,wm} ⊂ L2(Rd)associated to a Bloch frame {φ1, . . . , φm} are defined as
wa(x) :=(U−1φa
)(x) =
1
|Bb|
∫Bb
dk eik·xφa(k, x).
Localization of CWFs inposition space
−−−−−−−→BF transform
Smoothness of Blochframe in momentum space
∫Rd
〈x〉2s |wa,γ(x)|2dx ←→ ||φa||2Hs (Td∗,Hb
f )
Hence, by Sobolev theorem, for s > d/2 the existence of s-localized CWFs istopologically obstructed.
But, for 2 ≤ d ≤ 3, still there might exist 1-localized CWFs, i. e. 〈X 2〉w < +∞.
NO! In the topologicallynon-trivial case, does NOT
exist any system of1-localized composite
Wannier functions!
[St] Stein: Interpolation of linear operators. Trans. Am. Math. Soc. 83 (1956)
G. Panati La Sapienza The Localization-Topology Correspondence 20 / 33
Definition (CWFs): The composite Wannier functions {w1, . . . ,wm} ⊂ L2(Rd)associated to a Bloch frame {φ1, . . . , φm} are defined as
wa(x) :=(U−1φa
)(x) =
1
|Bb|
∫Bb
dk eik·xφa(k, x).
Localization of CWFs inposition space
−−−−−−−→BF transform
Smoothness of Blochframe in momentum space
∫Rd
〈x〉2s |wa,γ(x)|2dx ←→ ||φa||2Hs (Td∗,Hb
f )
Hence, by Sobolev theorem, for s > d/2 the existence of s-localized CWFs istopologically obstructed.
But, for 2 ≤ d ≤ 3, still there might exist 1-localized CWFs, i. e. 〈X 2〉w < +∞.
NO! In the topologicallynon-trivial case, does NOT
exist any system of1-localized composite
Wannier functions!
[St] Stein: Interpolation of linear operators. Trans. Am. Math. Soc. 83 (1956)
G. Panati La Sapienza The Localization-Topology Correspondence 20 / 33
Definition (CWFs): The composite Wannier functions {w1, . . . ,wm} ⊂ L2(Rd)associated to a Bloch frame {φ1, . . . , φm} are defined as
wa(x) :=(U−1φa
)(x) =
1
|Bb|
∫Bb
dk eik·xφa(k, x).
Localization of CWFs inposition space
−−−−−−−→BF transform
Smoothness of Blochframe in momentum space
∫Rd
〈x〉2s |wa,γ(x)|2dx ←→ ||φa||2Hs (Td∗,Hb
f )
Hence, by Sobolev theorem, for s > d/2 the existence of s-localized CWFs istopologically obstructed.
But, for 2 ≤ d ≤ 3, still there might exist 1-localized CWFs, i. e. 〈X 2〉w < +∞.
NO! In the topologicallynon-trivial case, does NOT
exist any system of1-localized composite
Wannier functions!
[St] Stein: Interpolation of linear operators. Trans. Am. Math. Soc. 83 (1956)
G. Panati La Sapienza The Localization-Topology Correspondence 20 / 33
Definition (CWFs): The composite Wannier functions {w1, . . . ,wm} ⊂ L2(Rd)associated to a Bloch frame {φ1, . . . , φm} are defined as
wa(x) :=(U−1φa
)(x) =
1
|Bb|
∫Bb
dk eik·xφa(k, x).
Localization of CWFs inposition space
−−−−−−−→BF transform
Smoothness of Blochframe in momentum space
∫Rd
〈x〉2s |wa,γ(x)|2dx ←→ ||φa||2Hs (Td∗,Hb
f )
Hence, by Sobolev theorem, for s > d/2 the existence of s-localized CWFs istopologically obstructed.
But, for 2 ≤ d ≤ 3, still there might exist 1-localized CWFs, i. e. 〈X 2〉w < +∞.
NO! In the topologicallynon-trivial case, does NOT
exist any system of1-localized composite
Wannier functions!
[St] Stein: Interpolation of linear operators. Trans. Am. Math. Soc. 83 (1956)
G. Panati La Sapienza The Localization-Topology Correspondence 20 / 33
Definition (CWFs): The composite Wannier functions {w1, . . . ,wm} ⊂ L2(Rd)associated to a Bloch frame {φ1, . . . , φm} are defined as
wa(x) :=(U−1φa
)(x) =
1
|Bb|
∫Bb
dk eik·xφa(k, x).
Localization of CWFs inposition space
−−−−−−−→BF transform
Smoothness of Blochframe in momentum space
∫Rd
〈x〉2s |wa,γ(x)|2dx ←→ ||φa||2Hs (Td∗,Hb
f )
Hence, by Sobolev theorem, for s > d/2 the existence of s-localized CWFs istopologically obstructed.
But, for 2 ≤ d ≤ 3, still there might exist 1-localized CWFs, i. e. 〈X 2〉w < +∞.
NO! In the topologicallynon-trivial case, does NOT
exist any system of1-localized composite
Wannier functions!
[St] Stein: Interpolation of linear operators. Trans. Am. Math. Soc. 83 (1956)
G. Panati La Sapienza The Localization-Topology Correspondence 20 / 33
The constructive theorem
Theorem 1 [Monaco, GP, Pisante, Teufel ’17]Assume d ≤ 3. Consider a magnetic periodic Schrodinger operator in L2(Rd) sati-sfying the previous assumptions (Kato smallness + commensurability + gap).Then we construct a Bloch frame in Hs(Td ;Hm) for every s < 1 and,correspondingly, a system of CWFs {wa,γ} such that
m∑a=1
∫Rd
〈x〉2s |wa,γ(x)|2dx < +∞ ∀γ ∈ Γ,∀s < 1.
G. Panati La Sapienza The Localization-Topology Correspondence 21 / 33
The Localization-Topology Correspondence
Theorem 2 [Monaco, GP, Pisante, Teufel ’17]Assume d ≤ 3. Consider a magnetic periodic Schrodinger operator in L2(Rd ) satisfying the
previous assumptions (Kato smallness + commensurability + gap).
The following statements are equivalent:
I Finite second moment: there exist CWFs {wa,γ} such that
m∑a=1
∫Rd
〈x〉2|wa,γ(x)|2dx < +∞ ∀γ ∈ Γ;
I Exponential localization: there exist CWFs {wa,γ} and α > 0 such that
m∑a=1
∫Rd
e2β|x||wa,γ(x)|2dx < +∞ ∀γ ∈ Γ, β ∈ [0, α);
I Trivial topology: the family {P∗(k)}k corresponds to a trivial Bloch bundle.
G. Panati La Sapienza The Localization-Topology Correspondence 22 / 33
Synopsis: symmetry, localization, transport, topology
G. Panati La Sapienza The Localization-Topology Correspondence 23 / 33
References
First column: existence of exponentially localized CWFs
[Ko] Kohn, W., Phys. Rev. 115 (1959) (m = 1, d = 1, even)[dC] des Cloizeaux, J., Phys. Rev. 135 (1964) (m = 1, any d , even)[NeNe1] Nenciu, A.; Nenciu, G., J. Phys. A 15 (1982) (any m, d = 1)[Ne1] Nenciu, G., Commun. Math. Phys. 91 (1983) (m = 1, any d)[HeSj] Helffer, Sjostrand, LNP (1989) (m = 1, any d)[Pa] G.P., Ann. Henri Poincare 8 (2007) (any m, d ≤ 3, abstract)[BPCM] Brouder Ch. et al., Phys. Rev. Lett. 98 (2007).(any m, d ≤ 3, abstract)[FMP] Fiorenza et al., Ann. Henri Poincare (2016) (any m, d ≤ 3, construct)[CHN] Cornean et al., Ann. Henri Poincare (2016) (any m, d ≤ 2, construct)[CLPS] Cances et al. : Phys. Rev B 95 (2017) (any m, d ≤ 3, algorithm)
Second column: power-law decay of CWFs in the topological phase
[Th] Thouless, J. Phys. C 17 (1984) (conjectured w(x) � |x |−2 for d = 2)[TV] Thonhauser, Vanderbilt, Phys. Rev. B (2006). (numerics on Haldane model)
Our result applies to all d ≤ 3 and is largely model-independent, since it coversboth continuous and tight-binding models.
G. Panati La Sapienza The Localization-Topology Correspondence 24 / 33
References
First column: existence of exponentially localized CWFs
[Ko] Kohn, W., Phys. Rev. 115 (1959) (m = 1, d = 1, even)[dC] des Cloizeaux, J., Phys. Rev. 135 (1964) (m = 1, any d , even)[NeNe1] Nenciu, A.; Nenciu, G., J. Phys. A 15 (1982) (any m, d = 1)[Ne1] Nenciu, G., Commun. Math. Phys. 91 (1983) (m = 1, any d)[HeSj] Helffer, Sjostrand, LNP (1989) (m = 1, any d)[Pa] G.P., Ann. Henri Poincare 8 (2007) (any m, d ≤ 3, abstract)[BPCM] Brouder Ch. et al., Phys. Rev. Lett. 98 (2007).(any m, d ≤ 3, abstract)[FMP] Fiorenza et al., Ann. Henri Poincare (2016) (any m, d ≤ 3, construct)[CHN] Cornean et al., Ann. Henri Poincare (2016) (any m, d ≤ 2, construct)[CLPS] Cances et al. : Phys. Rev B 95 (2017) (any m, d ≤ 3, algorithm)
Second column: power-law decay of CWFs in the topological phase
[Th] Thouless, J. Phys. C 17 (1984) (conjectured w(x) � |x |−2 for d = 2)[TV] Thonhauser, Vanderbilt, Phys. Rev. B (2006). (numerics on Haldane model)
Our result applies to all d ≤ 3 and is largely model-independent, since it coversboth continuous and tight-binding models.
G. Panati La Sapienza The Localization-Topology Correspondence 24 / 33
References
First column: existence of exponentially localized CWFs
[Ko] Kohn, W., Phys. Rev. 115 (1959) (m = 1, d = 1, even)[dC] des Cloizeaux, J., Phys. Rev. 135 (1964) (m = 1, any d , even)[NeNe1] Nenciu, A.; Nenciu, G., J. Phys. A 15 (1982) (any m, d = 1)[Ne1] Nenciu, G., Commun. Math. Phys. 91 (1983) (m = 1, any d)[HeSj] Helffer, Sjostrand, LNP (1989) (m = 1, any d)[Pa] G.P., Ann. Henri Poincare 8 (2007) (any m, d ≤ 3, abstract)[BPCM] Brouder Ch. et al., Phys. Rev. Lett. 98 (2007).(any m, d ≤ 3, abstract)[FMP] Fiorenza et al., Ann. Henri Poincare (2016) (any m, d ≤ 3, construct)[CHN] Cornean et al., Ann. Henri Poincare (2016) (any m, d ≤ 2, construct)[CLPS] Cances et al. : Phys. Rev B 95 (2017) (any m, d ≤ 3, algorithm)
Second column: power-law decay of CWFs in the topological phase
[Th] Thouless, J. Phys. C 17 (1984) (conjectured w(x) � |x |−2 for d = 2)[TV] Thonhauser, Vanderbilt, Phys. Rev. B (2006). (numerics on Haldane model)
Our result applies to all d ≤ 3 and is largely model-independent, since it coversboth continuous and tight-binding models.
G. Panati La Sapienza The Localization-Topology Correspondence 24 / 33
References
First column: existence of exponentially localized CWFs
[Ko] Kohn, W., Phys. Rev. 115 (1959) (m = 1, d = 1, even)[dC] des Cloizeaux, J., Phys. Rev. 135 (1964) (m = 1, any d , even)[NeNe1] Nenciu, A.; Nenciu, G., J. Phys. A 15 (1982) (any m, d = 1)[Ne1] Nenciu, G., Commun. Math. Phys. 91 (1983) (m = 1, any d)[HeSj] Helffer, Sjostrand, LNP (1989) (m = 1, any d)[Pa] G.P., Ann. Henri Poincare 8 (2007) (any m, d ≤ 3, abstract)[BPCM] Brouder Ch. et al., Phys. Rev. Lett. 98 (2007).(any m, d ≤ 3, abstract)[FMP] Fiorenza et al., Ann. Henri Poincare (2016) (any m, d ≤ 3, construct)[CHN] Cornean et al., Ann. Henri Poincare (2016) (any m, d ≤ 2, construct)[CLPS] Cances et al. : Phys. Rev B 95 (2017) (any m, d ≤ 3, algorithm)
Second column: power-law decay of CWFs in the topological phase
[Th] Thouless, J. Phys. C 17 (1984) (conjectured w(x) � |x |−2 for d = 2)[TV] Thonhauser, Vanderbilt, Phys. Rev. B (2006). (numerics on Haldane model)
Our result applies to all d ≤ 3 and is largely model-independent, since it coversboth continuous and tight-binding models.
G. Panati La Sapienza The Localization-Topology Correspondence 24 / 33
Grassmanian reinterpretation
Gm(H) :={P ∈ B(H) : P2 = P = P∗,TrP = m
}(Grassmann manifold)
Wm(H) := {J : Cm → H linear isometry} (Stiefel manifold),
Notice that J =∑m
a=1 |ψa〉 〈ea|, where {ψa} ⊂ H is an m-frame. There is a natural map
π : Wm(H)→ Gm(H) sending each m-frame Ψ = {ψ1, . . . , ψm} into the orthogonal projection
on its linear span, namely π : J 7→ JJ∗ =∑m
a=1 |ψa〉 〈ψa| .
At least formally, Wm(H) is a principal bundle over Gm(H) with projection π andfiber U(Cm). The data P and Φ correspond to a commutative diagram:
where P ∈ C∞(Td ;B(H)) and Φ ∈ Hs(Td ;Hm).
c1(P) =1
2πi
∫T2∗
TrH(P(k) [∂k1P(k), ∂k2P(k)]
)dk1 ∧ dk2
=1
2πi
∫T2∗
m∑a=1
2 Im 〈∂k1φa(k), ∂k2φa(k)〉H dk1 ∧ dk2.
Since the set of m-framesis not a linear space, theapproximation of a givenSobolev map by smooth
maps might betopologically obstructed.
G. Panati La Sapienza The Localization-Topology Correspondence 25 / 33
Grassmanian reinterpretation
Gm(H) :={P ∈ B(H) : P2 = P = P∗,TrP = m
}(Grassmann manifold)
Wm(H) := {J : Cm → H linear isometry} (Stiefel manifold),
Notice that J =∑m
a=1 |ψa〉 〈ea|, where {ψa} ⊂ H is an m-frame. There is a natural map
π : Wm(H)→ Gm(H) sending each m-frame Ψ = {ψ1, . . . , ψm} into the orthogonal projection
on its linear span, namely π : J 7→ JJ∗ =∑m
a=1 |ψa〉 〈ψa| .At least formally, Wm(H) is a principal bundle over Gm(H) with projection π andfiber U(Cm). The data P and Φ correspond to a commutative diagram:
where P ∈ C∞(Td ;B(H)) and Φ ∈ Hs(Td ;Hm).
c1(P) =1
2πi
∫T2∗
TrH(P(k) [∂k1P(k), ∂k2P(k)]
)dk1 ∧ dk2
=1
2πi
∫T2∗
m∑a=1
2 Im 〈∂k1φa(k), ∂k2φa(k)〉H dk1 ∧ dk2.
Since the set of m-framesis not a linear space, theapproximation of a givenSobolev map by smooth
maps might betopologically obstructed.
G. Panati La Sapienza The Localization-Topology Correspondence 25 / 33
Grassmanian reinterpretation
Gm(H) :={P ∈ B(H) : P2 = P = P∗,TrP = m
}(Grassmann manifold)
Wm(H) := {J : Cm → H linear isometry} (Stiefel manifold),
Notice that J =∑m
a=1 |ψa〉 〈ea|, where {ψa} ⊂ H is an m-frame. There is a natural map
π : Wm(H)→ Gm(H) sending each m-frame Ψ = {ψ1, . . . , ψm} into the orthogonal projection
on its linear span, namely π : J 7→ JJ∗ =∑m
a=1 |ψa〉 〈ψa| .At least formally, Wm(H) is a principal bundle over Gm(H) with projection π andfiber U(Cm). The data P and Φ correspond to a commutative diagram:
where P ∈ C∞(Td ;B(H)) and Φ ∈ Hs(Td ;Hm).
c1(P) =1
2πi
∫T2∗
TrH(P(k) [∂k1P(k), ∂k2P(k)]
)dk1 ∧ dk2
=1
2πi
∫T2∗
m∑a=1
2 Im 〈∂k1φa(k), ∂k2φa(k)〉H dk1 ∧ dk2.
Since the set of m-framesis not a linear space, theapproximation of a givenSobolev map by smooth
maps might betopologically obstructed.
G. Panati La Sapienza The Localization-Topology Correspondence 25 / 33
Grassmanian reinterpretation
Gm(H) :={P ∈ B(H) : P2 = P = P∗,TrP = m
}(Grassmann manifold)
Wm(H) := {J : Cm → H linear isometry} (Stiefel manifold),
Notice that J =∑m
a=1 |ψa〉 〈ea|, where {ψa} ⊂ H is an m-frame. There is a natural map
π : Wm(H)→ Gm(H) sending each m-frame Ψ = {ψ1, . . . , ψm} into the orthogonal projection
on its linear span, namely π : J 7→ JJ∗ =∑m
a=1 |ψa〉 〈ψa| .At least formally, Wm(H) is a principal bundle over Gm(H) with projection π andfiber U(Cm). The data P and Φ correspond to a commutative diagram:
where P ∈ C∞(Td ;B(H)) and Φ ∈ Hs(Td ;Hm).
c1(P) =1
2πi
∫T2∗
TrH(P(k) [∂k1P(k), ∂k2P(k)]
)dk1 ∧ dk2
=1
2πi
∫T2∗
m∑a=1
2 Im 〈∂k1φa(k), ∂k2φa(k)〉H dk1 ∧ dk2.
Since the set of m-framesis not a linear space, theapproximation of a givenSobolev map by smooth
maps might betopologically obstructed.
G. Panati La Sapienza The Localization-Topology Correspondence 25 / 33
Grassmanian reinterpretation
Gm(H) :={P ∈ B(H) : P2 = P = P∗,TrP = m
}(Grassmann manifold)
Wm(H) := {J : Cm → H linear isometry} (Stiefel manifold),
Notice that J =∑m
a=1 |ψa〉 〈ea|, where {ψa} ⊂ H is an m-frame. There is a natural map
π : Wm(H)→ Gm(H) sending each m-frame Ψ = {ψ1, . . . , ψm} into the orthogonal projection
on its linear span, namely π : J 7→ JJ∗ =∑m
a=1 |ψa〉 〈ψa| .At least formally, Wm(H) is a principal bundle over Gm(H) with projection π andfiber U(Cm). The data P and Φ correspond to a commutative diagram:
where P ∈ C∞(Td ;B(H)) and Φ ∈ Hs(Td ;Hm).
c1(P) =1
2πi
∫T2∗
TrH(P(k) [∂k1P(k), ∂k2P(k)]
)dk1 ∧ dk2
=1
2πi
∫T2∗
m∑a=1
2 Im 〈∂k1φa(k), ∂k2φa(k)〉H dk1 ∧ dk2.
Since the set of m-framesis not a linear space, theapproximation of a givenSobolev map by smooth
maps might betopologically obstructed.
G. Panati La Sapienza The Localization-Topology Correspondence 25 / 33
Approximation of a given Sobolev map
Theorem [Hang & Lin]Let 2 ≤ d ≤ 3. Consider a compact, boundaryless, smooth submanifold M ⊂ Rν .If d = 3, assume moreover that the homotopy group π2(M) is trivial.Then, every Sobolev map Ψ ∈ H1(Td ,M) can be approximated by a sequence
{Ψ(`)}`∈N ⊂ C∞(Td ,M) such that Ψ(`) H1
−→ Ψ as `→∞.
The result can be applied to the (finite-dimensional) Stiefel manifoldWm(Cn), using that π2(Wm(Cn)) = 0 whenever n ≥ m + 2 (here Cn isregarded as a Galerkin truncation of H).One has to reduce to a finite-dimensional space Vn ' Cn
One has to reduce covariance to periodicity, while staying in a k-independentfiber Hilber space [CHN]In our specific case, there is no obstruction to construct a Bloch frame withregularity Hs(Td ;Hm) for every s < 1, essentially via parallel transport[construction in the paper].
[HL] Hang, F.; Lin, F.H. : Topology of Sobolev mappings. II. Acta Math. 2003.[CHN] Cornean, H.; Herbst, I., Nenciu, G. : Ann. H. Poincare. 2016.
G. Panati La Sapienza The Localization-Topology Correspondence 26 / 33
Approximation of a given Sobolev map
Theorem [Hang & Lin]Let 2 ≤ d ≤ 3. Consider a compact, boundaryless, smooth submanifold M ⊂ Rν .If d = 3, assume moreover that the homotopy group π2(M) is trivial.Then, every Sobolev map Ψ ∈ H1(Td ,M) can be approximated by a sequence
{Ψ(`)}`∈N ⊂ C∞(Td ,M) such that Ψ(`) H1
−→ Ψ as `→∞.
The result can be applied to the (finite-dimensional) Stiefel manifoldWm(Cn), using that π2(Wm(Cn)) = 0 whenever n ≥ m + 2 (here Cn isregarded as a Galerkin truncation of H).
One has to reduce to a finite-dimensional space Vn ' Cn
One has to reduce covariance to periodicity, while staying in a k-independentfiber Hilber space [CHN]In our specific case, there is no obstruction to construct a Bloch frame withregularity Hs(Td ;Hm) for every s < 1, essentially via parallel transport[construction in the paper].
[HL] Hang, F.; Lin, F.H. : Topology of Sobolev mappings. II. Acta Math. 2003.[CHN] Cornean, H.; Herbst, I., Nenciu, G. : Ann. H. Poincare. 2016.
G. Panati La Sapienza The Localization-Topology Correspondence 26 / 33
Approximation of a given Sobolev map
Theorem [Hang & Lin]Let 2 ≤ d ≤ 3. Consider a compact, boundaryless, smooth submanifold M ⊂ Rν .If d = 3, assume moreover that the homotopy group π2(M) is trivial.Then, every Sobolev map Ψ ∈ H1(Td ,M) can be approximated by a sequence
{Ψ(`)}`∈N ⊂ C∞(Td ,M) such that Ψ(`) H1
−→ Ψ as `→∞.
The result can be applied to the (finite-dimensional) Stiefel manifoldWm(Cn), using that π2(Wm(Cn)) = 0 whenever n ≥ m + 2 (here Cn isregarded as a Galerkin truncation of H).One has to reduce to a finite-dimensional space Vn ' Cn
One has to reduce covariance to periodicity, while staying in a k-independentfiber Hilber space [CHN]In our specific case, there is no obstruction to construct a Bloch frame withregularity Hs(Td ;Hm) for every s < 1, essentially via parallel transport[construction in the paper].
[HL] Hang, F.; Lin, F.H. : Topology of Sobolev mappings. II. Acta Math. 2003.[CHN] Cornean, H.; Herbst, I., Nenciu, G. : Ann. H. Poincare. 2016.
G. Panati La Sapienza The Localization-Topology Correspondence 26 / 33
Approximation of a given Sobolev map
Theorem [Hang & Lin]Let 2 ≤ d ≤ 3. Consider a compact, boundaryless, smooth submanifold M ⊂ Rν .If d = 3, assume moreover that the homotopy group π2(M) is trivial.Then, every Sobolev map Ψ ∈ H1(Td ,M) can be approximated by a sequence
{Ψ(`)}`∈N ⊂ C∞(Td ,M) such that Ψ(`) H1
−→ Ψ as `→∞.
The result can be applied to the (finite-dimensional) Stiefel manifoldWm(Cn), using that π2(Wm(Cn)) = 0 whenever n ≥ m + 2 (here Cn isregarded as a Galerkin truncation of H).One has to reduce to a finite-dimensional space Vn ' Cn
One has to reduce covariance to periodicity, while staying in a k-independentfiber Hilber space [CHN]
In our specific case, there is no obstruction to construct a Bloch frame withregularity Hs(Td ;Hm) for every s < 1, essentially via parallel transport[construction in the paper].
[HL] Hang, F.; Lin, F.H. : Topology of Sobolev mappings. II. Acta Math. 2003.[CHN] Cornean, H.; Herbst, I., Nenciu, G. : Ann. H. Poincare. 2016.
G. Panati La Sapienza The Localization-Topology Correspondence 26 / 33
Approximation of a given Sobolev map
Theorem [Hang & Lin]Let 2 ≤ d ≤ 3. Consider a compact, boundaryless, smooth submanifold M ⊂ Rν .If d = 3, assume moreover that the homotopy group π2(M) is trivial.Then, every Sobolev map Ψ ∈ H1(Td ,M) can be approximated by a sequence
{Ψ(`)}`∈N ⊂ C∞(Td ,M) such that Ψ(`) H1
−→ Ψ as `→∞.
The result can be applied to the (finite-dimensional) Stiefel manifoldWm(Cn), using that π2(Wm(Cn)) = 0 whenever n ≥ m + 2 (here Cn isregarded as a Galerkin truncation of H).One has to reduce to a finite-dimensional space Vn ' Cn
One has to reduce covariance to periodicity, while staying in a k-independentfiber Hilber space [CHN]In our specific case, there is no obstruction to construct a Bloch frame withregularity Hs(Td ;Hm) for every s < 1, essentially via parallel transport[construction in the paper].
[HL] Hang, F.; Lin, F.H. : Topology of Sobolev mappings. II. Acta Math. 2003.[CHN] Cornean, H.; Herbst, I., Nenciu, G. : Ann. H. Poincare. 2016.
G. Panati La Sapienza The Localization-Topology Correspondence 26 / 33
Part II
The Localization-Topology Correspondence:
the non-periodic case
Work in progress with G. Marcelli and M. Moscolari
G. Panati La Sapienza The Localization-Topology Correspondence 27 / 33
A modified paradigm
I Recall the periodic TKNN paradigm:
σ(Kubo)xy︸ ︷︷ ︸
Transport
= − i
(2π)2
∫T2
Tr(P(k)
[∂k1P(k), ∂k2P(k)
])dk =
1
2πC1(EP)︸ ︷︷ ︸Topology
I In a non-periodic setting, the decomposition {P(k)}k∈Td over the Brillouintorus makes no sense, and so ∂kj and the integral should be reinterpreted.
I Inspired by [AS2], we can write
σ(Kubo)xy = T
(iP[[X1,P], [X2,P]
])
=:1
2πC1(P)︸ ︷︷ ︸
NC Topology?
where the trace per unit volume is T (A) = limΛn↗Rd |Λn|−1 Tr(χΛn AχΛn).
I Analogy with the NCG approach to QHE [Co, Be, BES, Ke, . . . ]
C1(p) ' τ(ip[∂1(p), ∂2(p)
])for p a projector in the rotation C∗-algebra ' NC torus. Here C1(p) ∈ Z.
Questions for experts in NCG:
Is T (·) a tracial state over aC∗-subalgebra of B(L2(Rd))?
Which one? NCG?
Question we adressed:
Is there any relation betweenC1(P) = 0 and the existence
of a well-localized GWB??
G. Panati La Sapienza The Localization-Topology Correspondence 28 / 33
A modified paradigm
I Recall the periodic TKNN paradigm:
σ(Kubo)xy︸ ︷︷ ︸
Transport
= − i
(2π)2
∫T2
Tr(P(k)
[∂k1P(k), ∂k2P(k)
])dk =
1
2πC1(EP)︸ ︷︷ ︸Topology
I In a non-periodic setting, the decomposition {P(k)}k∈Td over the Brillouintorus makes no sense, and so ∂kj and the integral should be reinterpreted.
I Inspired by [AS2], we can write
σ(Kubo)xy = T
(iP[[X1,P], [X2,P]
])
=:1
2πC1(P)︸ ︷︷ ︸
NC Topology?
where the trace per unit volume is T (A) = limΛn↗Rd |Λn|−1 Tr(χΛn AχΛn).
I Analogy with the NCG approach to QHE [Co, Be, BES, Ke, . . . ]
C1(p) ' τ(ip[∂1(p), ∂2(p)
])for p a projector in the rotation C∗-algebra ' NC torus. Here C1(p) ∈ Z.
Questions for experts in NCG:
Is T (·) a tracial state over aC∗-subalgebra of B(L2(Rd))?
Which one? NCG?
Question we adressed:
Is there any relation betweenC1(P) = 0 and the existence
of a well-localized GWB??
G. Panati La Sapienza The Localization-Topology Correspondence 28 / 33
A modified paradigm
I Recall the periodic TKNN paradigm:
σ(Kubo)xy︸ ︷︷ ︸
Transport
= − i
(2π)2
∫T2
Tr(P(k)
[∂k1P(k), ∂k2P(k)
])dk =
1
2πC1(EP)︸ ︷︷ ︸Topology
I In a non-periodic setting, the decomposition {P(k)}k∈Td over the Brillouintorus makes no sense, and so ∂kj and the integral should be reinterpreted.
I Inspired by [AS2], we can write
σ(Kubo)xy = T
(iP[[X1,P], [X2,P]
])
=:1
2πC1(P)︸ ︷︷ ︸
NC Topology?
where the trace per unit volume is T (A) = limΛn↗Rd |Λn|−1 Tr(χΛn AχΛn).
I Analogy with the NCG approach to QHE [Co, Be, BES, Ke, . . . ]
C1(p) ' τ(ip[∂1(p), ∂2(p)
])for p a projector in the rotation C∗-algebra ' NC torus. Here C1(p) ∈ Z.
Questions for experts in NCG:
Is T (·) a tracial state over aC∗-subalgebra of B(L2(Rd))?
Which one? NCG?
Question we adressed:
Is there any relation betweenC1(P) = 0 and the existence
of a well-localized GWB??
G. Panati La Sapienza The Localization-Topology Correspondence 28 / 33
A modified paradigm
I Recall the periodic TKNN paradigm:
σ(Kubo)xy︸ ︷︷ ︸
Transport
= − i
(2π)2
∫T2
Tr(P(k)
[∂k1P(k), ∂k2P(k)
])dk =
1
2πC1(EP)︸ ︷︷ ︸Topology
I In a non-periodic setting, the decomposition {P(k)}k∈Td over the Brillouintorus makes no sense, and so ∂kj and the integral should be reinterpreted.
I Inspired by [AS2], we can write
σ(Kubo)xy = T
(iP[[X1,P], [X2,P]
])=:
1
2πC1(P)︸ ︷︷ ︸
NC Topology?
where the trace per unit volume is T (A) = limΛn↗Rd |Λn|−1 Tr(χΛn AχΛn).
I Analogy with the NCG approach to QHE [Co, Be, BES, Ke, . . . ]
C1(p) ' τ(ip[∂1(p), ∂2(p)
])for p a projector in the rotation C∗-algebra ' NC torus. Here C1(p) ∈ Z.
Questions for experts in NCG:
Is T (·) a tracial state over aC∗-subalgebra of B(L2(Rd))?
Which one? NCG?
Question we adressed:
Is there any relation betweenC1(P) = 0 and the existence
of a well-localized GWB??
G. Panati La Sapienza The Localization-Topology Correspondence 28 / 33
A modified paradigm
I Recall the periodic TKNN paradigm:
σ(Kubo)xy︸ ︷︷ ︸
Transport
= − i
(2π)2
∫T2
Tr(P(k)
[∂k1P(k), ∂k2P(k)
])dk =
1
2πC1(EP)︸ ︷︷ ︸Topology
I In a non-periodic setting, the decomposition {P(k)}k∈Td over the Brillouintorus makes no sense, and so ∂kj and the integral should be reinterpreted.
I Inspired by [AS2], we can write
σ(Kubo)xy = T
(iP[[X1,P], [X2,P]
])=:
1
2πC1(P)︸ ︷︷ ︸
NC Topology?
where the trace per unit volume is T (A) = limΛn↗Rd |Λn|−1 Tr(χΛn AχΛn).
I Analogy with the NCG approach to QHE [Co, Be, BES, Ke, . . . ]
C1(p) ' τ(ip[∂1(p), ∂2(p)
])for p a projector in the rotation C∗-algebra ' NC torus. Here C1(p) ∈ Z.
Questions for experts in NCG:
Is T (·) a tracial state over aC∗-subalgebra of B(L2(Rd))?
Which one? NCG?
Question we adressed:
Is there any relation betweenC1(P) = 0 and the existence
of a well-localized GWB??
G. Panati La Sapienza The Localization-Topology Correspondence 28 / 33
A modified paradigm
I Recall the periodic TKNN paradigm:
σ(Kubo)xy︸ ︷︷ ︸
Transport
= − i
(2π)2
∫T2
Tr(P(k)
[∂k1P(k), ∂k2P(k)
])dk =
1
2πC1(EP)︸ ︷︷ ︸Topology
I In a non-periodic setting, the decomposition {P(k)}k∈Td over the Brillouintorus makes no sense, and so ∂kj and the integral should be reinterpreted.
I Inspired by [AS2], we can write
σ(Kubo)xy = T
(iP[[X1,P], [X2,P]
])=:
1
2πC1(P)︸ ︷︷ ︸
NC Topology?
where the trace per unit volume is T (A) = limΛn↗Rd |Λn|−1 Tr(χΛn AχΛn).
I Analogy with the NCG approach to QHE [Co, Be, BES, Ke, . . . ]
C1(p) ' τ(ip[∂1(p), ∂2(p)
])for p a projector in the rotation C∗-algebra ' NC torus. Here C1(p) ∈ Z.
Questions for experts in NCG:
Is T (·) a tracial state over aC∗-subalgebra of B(L2(Rd))?
Which one? NCG?
Question we adressed:
Is there any relation betweenC1(P) = 0 and the existence
of a well-localized GWB??
G. Panati La Sapienza The Localization-Topology Correspondence 28 / 33
A modified paradigm
I Recall the periodic TKNN paradigm:
σ(Kubo)xy︸ ︷︷ ︸
Transport
= − i
(2π)2
∫T2
Tr(P(k)
[∂k1P(k), ∂k2P(k)
])dk =
1
2πC1(EP)︸ ︷︷ ︸Topology
I In a non-periodic setting, the decomposition {P(k)}k∈Td over the Brillouintorus makes no sense, and so ∂kj and the integral should be reinterpreted.
I Inspired by [AS2], we can write
σ(Kubo)xy = T
(iP[[X1,P], [X2,P]
])=:
1
2πC1(P)︸ ︷︷ ︸
NC Topology?
where the trace per unit volume is T (A) = limΛn↗Rd |Λn|−1 Tr(χΛn AχΛn).
I Analogy with the NCG approach to QHE [Co, Be, BES, Ke, . . . ]
C1(p) ' τ(ip[∂1(p), ∂2(p)
])for p a projector in the rotation C∗-algebra ' NC torus. Here C1(p) ∈ Z.
Questions for experts in NCG:
Is T (·) a tracial state over aC∗-subalgebra of B(L2(Rd))?
Which one? NCG?
Question we adressed:
Is there any relation betweenC1(P) = 0 and the existence
of a well-localized GWB??
G. Panati La Sapienza The Localization-Topology Correspondence 28 / 33
Definition (Generalized Wannier basis) (compare with [NN93])
An orthogonal projector P acting in L2(R2) admits a G-localized generalizedWannier basis (GWB) if there exist:
(i) a Delone set Γ ⊆ R2, i. e. a discrete set such that ∃ 0 < r < R <∞ s.t.
(a) ∀x ∈ R2 there is at most one element of Γ in the ball of radius r centred in x(the set has no accumulation points);
(b) ∀x ∈ R2 there is at least one element of Γ in the ball of radius R centred in x(the set is not sparse);
(ii) a localization function G (typically G (x) = (1 + |x |2)s/2 for some s ≥ 1), aconstant M > 0 independent of γ ∈ Γ and an orthonormal basis of RanP,{ψγ,a}γ∈Γ,1≤a≤m(γ)<∞ with m(γ) ≤ m∗ ∀γ ∈ Γ, satisfying∫
R2
G (|x − γ|)2|ψγ,a(x)|2 dx ≤ M.
We call each ψγ,a a generalized Wannier function (GWF).
[NN93] Gh. Nenciu, A. Nenciu Phys. Rev. B 47 (1993)
G. Panati La Sapienza The Localization-Topology Correspondence 29 / 33
Definition (Generalized Wannier basis) (compare with [NN93])
An orthogonal projector P acting in L2(R2) admits a G-localized generalizedWannier basis (GWB) if there exist:
(i) a Delone set Γ ⊆ R2, i. e. a discrete set such that ∃ 0 < r < R <∞ s.t.
(a) ∀x ∈ R2 there is at most one element of Γ in the ball of radius r centred in x(the set has no accumulation points);
(b) ∀x ∈ R2 there is at least one element of Γ in the ball of radius R centred in x(the set is not sparse);
(ii) a localization function G (typically G (x) = (1 + |x |2)s/2 for some s ≥ 1), aconstant M > 0 independent of γ ∈ Γ and an orthonormal basis of RanP,{ψγ,a}γ∈Γ,1≤a≤m(γ)<∞ with m(γ) ≤ m∗ ∀γ ∈ Γ, satisfying∫
R2
G (|x − γ|)2|ψγ,a(x)|2 dx ≤ M.
We call each ψγ,a a generalized Wannier function (GWF).
[NN93] Gh. Nenciu, A. Nenciu Phys. Rev. B 47 (1993)
G. Panati La Sapienza The Localization-Topology Correspondence 29 / 33
Definition (Generalized Wannier basis) (compare with [NN93])
An orthogonal projector P acting in L2(R2) admits a G-localized generalizedWannier basis (GWB) if there exist:
(i) a Delone set Γ ⊆ R2, i. e. a discrete set such that ∃ 0 < r < R <∞ s.t.
(a) ∀x ∈ R2 there is at most one element of Γ in the ball of radius r centred in x(the set has no accumulation points);
(b) ∀x ∈ R2 there is at least one element of Γ in the ball of radius R centred in x(the set is not sparse);
(ii) a localization function G (typically G (x) = (1 + |x |2)s/2 for some s ≥ 1), aconstant M > 0 independent of γ ∈ Γ and an orthonormal basis of RanP,{ψγ,a}γ∈Γ,1≤a≤m(γ)<∞ with m(γ) ≤ m∗ ∀γ ∈ Γ, satisfying∫
R2
G (|x − γ|)2|ψγ,a(x)|2 dx ≤ M.
We call each ψγ,a a generalized Wannier function (GWF).
[NN93] Gh. Nenciu, A. Nenciu Phys. Rev. B 47 (1993)
G. Panati La Sapienza The Localization-Topology Correspondence 29 / 33
Localization implies Chern triviality
Theorem - work in progress [Marcelli, Moscolari, GP]
Let Pµ be the Fermi projector of a reasonable Schrodinger operator in L2(R2).
Suppose that Pµ admits a generalized Wannier basis, {ψγ,a}γ∈Γ,1≤a≤m(γ)<m∗ , whichis s∗-localized in the sense∫
R2
(1 + |x − γ|2)s∗ |ψγ,a(x)|2 dx ≤ M.
Then, if s∗ ≥ 4 [provisional hypothesis], one has that
T(iPµ[[X1,Pµ], [X2,Pµ]
])= 0.
Clearly, the optimalstatement would befor s∗ = 1, as in the
periodic case.Technical difficulties.d = 1 in [NN93] it is proved that an exponentially localized GWB exists under
general hypothesis (Γ discrete): for d = 1, PµXPµ has discrete spectrum [!!!],and a GWB is provided by its eigenfunctions
{ψγ,a} γ ∈ σdisc(PµXPµ) =: Γ, a ∈ {1, . . . ,m(γ)} .
G. Panati La Sapienza The Localization-Topology Correspondence 30 / 33
Localization implies Chern triviality
Theorem - work in progress [Marcelli, Moscolari, GP]
Let Pµ be the Fermi projector of a reasonable Schrodinger operator in L2(R2).
Suppose that Pµ admits a generalized Wannier basis, {ψγ,a}γ∈Γ,1≤a≤m(γ)<m∗ , whichis s∗-localized in the sense∫
R2
(1 + |x − γ|2)s∗ |ψγ,a(x)|2 dx ≤ M.
Then, if s∗ ≥ 4 [provisional hypothesis], one has that
T(iPµ[[X1,Pµ], [X2,Pµ]
])= 0.
Clearly, the optimalstatement would befor s∗ = 1, as in the
periodic case.Technical difficulties.d = 1 in [NN93] it is proved that an exponentially localized GWB exists under
general hypothesis (Γ discrete): for d = 1, PµXPµ has discrete spectrum [!!!],and a GWB is provided by its eigenfunctions
{ψγ,a} γ ∈ σdisc(PµXPµ) =: Γ, a ∈ {1, . . . ,m(γ)} .
G. Panati La Sapienza The Localization-Topology Correspondence 30 / 33
Localization implies Chern triviality
Theorem - work in progress [Marcelli, Moscolari, GP]
Let Pµ be the Fermi projector of a reasonable Schrodinger operator in L2(R2).
Suppose that Pµ admits a generalized Wannier basis, {ψγ,a}γ∈Γ,1≤a≤m(γ)<m∗ , whichis s∗-localized in the sense∫
R2
(1 + |x − γ|2)s∗ |ψγ,a(x)|2 dx ≤ M.
Then, if s∗ ≥ 4 [provisional hypothesis], one has that
T(iPµ[[X1,Pµ], [X2,Pµ]
])= 0.
Clearly, the optimalstatement would befor s∗ = 1, as in the
periodic case.Technical difficulties.
d = 1 in [NN93] it is proved that an exponentially localized GWB exists undergeneral hypothesis (Γ discrete): for d = 1, PµXPµ has discrete spectrum [!!!],and a GWB is provided by its eigenfunctions
{ψγ,a} γ ∈ σdisc(PµXPµ) =: Γ, a ∈ {1, . . . ,m(γ)} .
G. Panati La Sapienza The Localization-Topology Correspondence 30 / 33
Localization implies Chern triviality
Theorem - work in progress [Marcelli, Moscolari, GP]
Let Pµ be the Fermi projector of a reasonable Schrodinger operator in L2(R2).
Suppose that Pµ admits a generalized Wannier basis, {ψγ,a}γ∈Γ,1≤a≤m(γ)<m∗ , whichis s∗-localized in the sense∫
R2
(1 + |x − γ|2)s∗ |ψγ,a(x)|2 dx ≤ M.
Then, if s∗ ≥ 4 [provisional hypothesis], one has that
T(iPµ[[X1,Pµ], [X2,Pµ]
])= 0.
Clearly, the optimalstatement would befor s∗ = 1, as in the
periodic case.Technical difficulties.
d = 1 in [NN93] it is proved that an exponentially localized GWB exists undergeneral hypothesis (Γ discrete): for d = 1, PµXPµ has discrete spectrum [!!!],and a GWB is provided by its eigenfunctions
{ψγ,a} γ ∈ σdisc(PµXPµ) =: Γ, a ∈ {1, . . . ,m(γ)} .
G. Panati La Sapienza The Localization-Topology Correspondence 30 / 33
A simple but relevant observation
I Let Xj := Pµ Xj Pµ be the reduced position operator. Then, by simple algebra
Pµ[[X1,Pµ], [X2,Pµ]
]=[X1, X2
]. (1)
If T (·) were cyclic, one would conclude that C1(P) is always zero.(Hall transport would be always forbidden!).
I Luckily for transport theory, T (·) is NOT cyclic in general.Cyclicity is recovered if it happens that
Pµ =∑γ,a
|ψγ,a〉 〈ψγ,a| (2)
where {ψγ,a} is a s-localized GWB, with s sufficiently large.
I Indeed, in this case the series obtained by plugging (2) in (1) convergesabsolutely, hence the series can be conveniently rearranged to obtaincancelations of clusters of terms, yielding C1(P) = 0.
I Boring details. Our estimates are not optimal.
G. Panati La Sapienza The Localization-Topology Correspondence 31 / 33
A simple but relevant observation
I Let Xj := Pµ Xj Pµ be the reduced position operator. Then, by simple algebra
Pµ[[X1,Pµ], [X2,Pµ]
]=[X1, X2
]. (1)
If T (·) were cyclic, one would conclude that C1(P) is always zero.(Hall transport would be always forbidden!).
I Luckily for transport theory, T (·) is NOT cyclic in general.Cyclicity is recovered if it happens that
Pµ =∑γ,a
|ψγ,a〉 〈ψγ,a| (2)
where {ψγ,a} is a s-localized GWB, with s sufficiently large.
I Indeed, in this case the series obtained by plugging (2) in (1) convergesabsolutely, hence the series can be conveniently rearranged to obtaincancelations of clusters of terms, yielding C1(P) = 0.
I Boring details. Our estimates are not optimal.
G. Panati La Sapienza The Localization-Topology Correspondence 31 / 33
A simple but relevant observation
I Let Xj := Pµ Xj Pµ be the reduced position operator. Then, by simple algebra
Pµ[[X1,Pµ], [X2,Pµ]
]=[X1, X2
]. (1)
If T (·) were cyclic, one would conclude that C1(P) is always zero.(Hall transport would be always forbidden!).
I Luckily for transport theory, T (·) is NOT cyclic in general.Cyclicity is recovered if it happens that
Pµ =∑γ,a
|ψγ,a〉 〈ψγ,a| (2)
where {ψγ,a} is a s-localized GWB, with s sufficiently large.
I Indeed, in this case the series obtained by plugging (2) in (1) convergesabsolutely, hence the series can be conveniently rearranged to obtaincancelations of clusters of terms, yielding C1(P) = 0.
I Boring details. Our estimates are not optimal.
G. Panati La Sapienza The Localization-Topology Correspondence 31 / 33
A simple but relevant observation
I Let Xj := Pµ Xj Pµ be the reduced position operator. Then, by simple algebra
Pµ[[X1,Pµ], [X2,Pµ]
]=[X1, X2
]. (1)
If T (·) were cyclic, one would conclude that C1(P) is always zero.(Hall transport would be always forbidden!).
I Luckily for transport theory, T (·) is NOT cyclic in general.Cyclicity is recovered if it happens that
Pµ =∑γ,a
|ψγ,a〉 〈ψγ,a| (2)
where {ψγ,a} is a s-localized GWB, with s sufficiently large.
I Indeed, in this case the series obtained by plugging (2) in (1) convergesabsolutely, hence the series can be conveniently rearranged to obtaincancelations of clusters of terms, yielding C1(P) = 0.
I Boring details. Our estimates are not optimal.
G. Panati La Sapienza The Localization-Topology Correspondence 31 / 33
A simple but relevant observation
I Let Xj := Pµ Xj Pµ be the reduced position operator. Then, by simple algebra
Pµ[[X1,Pµ], [X2,Pµ]
]=[X1, X2
]. (1)
If T (·) were cyclic, one would conclude that C1(P) is always zero.(Hall transport would be always forbidden!).
I Luckily for transport theory, T (·) is NOT cyclic in general.Cyclicity is recovered if it happens that
Pµ =∑γ,a
|ψγ,a〉 〈ψγ,a| (2)
where {ψγ,a} is a s-localized GWB, with s sufficiently large.
I Indeed, in this case the series obtained by plugging (2) in (1) convergesabsolutely, hence the series can be conveniently rearranged to obtaincancelations of clusters of terms, yielding C1(P) = 0.
I Boring details. Our estimates are not optimal.
G. Panati La Sapienza The Localization-Topology Correspondence 31 / 33
Thank you for your attention!!
G. Panati La Sapienza The Localization-Topology Correspondence 32 / 33
Synopsis: symmetry, localization, transport, topology
G. Panati La Sapienza The Localization-Topology Correspondence 33 / 33